LARGE AND SMALL GROUP HOMOLOGY

Transcription

1 LARGE AND SMALL GROUP HOMOLOGY MICHAEL BRUNNBAUER AND BERNHARD HANKE ABSTRACT. For several instances of metric largeness like enlargeability or having hyperspherical universal covers, we construct non-large vector subspaces in the rational homology of finitely generated groups. The functorial properties of this construction imply that the corresponding largeness properties of closed manifolds depend only on the image of their fundamental classes under the classifying map. This is applied to construct examples of essential manifolds whose universal covers are not hyperspherical, thus answering a question of Gromov (1986), and, more generally, essential manifolds which are not enlargeable. 1. INTRODUCTION Gromov and others [11, 14, 15] introduced notions of largeness for Riemannian manifolds. These include enlargeability and having hypereuclidean or hyperspherical universal covers, and universal covers with infinite filling radii. While precise definitions are given later in this paper, we point out that in spite of their reference to Riemannian metrics these properties are independent of the chosen metrics in the case of closed manifolds. In this paper we will elaborate on the topological-homological nature of several largeness properties of closed manifolds and more generally of homology classes of simplicial complexes with finitely generated fundamental groups. To explain some of our results in greater detail, let us here recall the definition of enlargeability (cf. [15]). In this paper all manifolds are assumed to be smooth and connected unless otherwise stated. Definition 1.1. Let M be a closed orientable manifold of dimension n, and let g be a Riemannian metric on M. Then (M, g) is called enlargeable, if for every ε > 0 there is a Riemannian cover M ε M and an ε-contracting (i. e. ε-lipschitz) map M ε S n to the n-dimensional unit sphere which is constant outside a compact set and of nonzero degree. On closed manifolds all Riemannian metrics are in bi-lipschitz correspondence and hence the described property does not depend on the particular choice of g. However, it is important that g remains fixed for the different choices of ε. Examples of enlargeable manifolds include tori and more generally manifolds admitting Riemannian metrics of nonpositive sectional curvature. Definition 1.2 ([9]). Let M be a closed oriented manifold with fundamental class [M], and let Φ : M Bπ 1 (M) be the classifying map of the universal cover. We call M essential if Φ [M] 0 H (Bπ 1 (M); Q). Date: March 31, Mathematics Subject Classification. Primary 53C23; Secondary 20J06. Key words and phrases. metric largeness, group homology. 1

2 In [19] it was shown by index theoretic methods that enlargeable manifolds are essential, if the cover M ε M can always be assumed to be finite. Relying on ideas from coarse geometry, [18, Corollary 1.3] states essentialness for manifolds with hyperspherical universal covers (these are manifolds where M ε may always be chosen as the universal cover). Elaborating on the methods from [19], essentialness of all enlargeable manifolds was proven in [20]. Extending the results of [18, 19, 20] we will show that manifolds which are either enlargeable or have universal covers which are (coarsely) hyperspherical, (coarsely) hypereuclidean or macroscopically large (these notions will be defined in Section 2) are essential. Our approach is independent of index theory or coarse geometry (unless the largeness condition under consideration refers to coarse notions). We will moreover prove that each of these largeness properties depends only on the image Φ [M] H (Bπ 1 (M); Q) of the fundamental class under the classifying map. This property may be called homological invariance of largeness. Both conclusions are implied by the following theorem which will be proved in Section 3 of our paper. Theorem 1.3. Let Γ be a finitely generated discrete group and let n be a natural number. Let P denote one of the properties of being enlargeable or having a universal cover which is (coarsely) hyperspherical, (coarsely) hypereuclidean or macroscopically large, there is a rational vector subspace Hn sm(p) (BΓ; Q) H n (BΓ; Q) of non-large (i. e. small) classes with respect to P. In the case when Γ is finitely presentable, the following holds: If M is a closed oriented n-dimensional manifold with fundamental group Γ and classifying map Φ : M BΓ, then Φ [M] Hn sm(p) (BΓ; Q) if and only if M is not large in the respective sense. In Definition 3.1 we will introduce largeness properties in general for homology classes of connected simplicial complexes with finitely generated fundamental groups by adapting the classical definitions accordingly. The hard part of this approach lies in the verification that the required properties depend on the given homology class only. In this respect homological invariance of largeness is built into the definition right away. Once this has been achieved it will be easy to show that the classes which are small, i. e. not large in the respective sense, enjoy the nice algebraic property of forming a vector subspace, see Theorem 3.6. If the simplicial complex in question is the classifying space of a finitely generated group, this approach emphasizes our point of view that the largeness properties in Theorem 1.3 should be regarded as metric properties of finitely generated groups (respectivelly their rational homology) much like the quasi-isometry type of the word metric itself. Homological invariance of largeness for the classical case of closed manifolds is a simple consequence of the functorial properties for large homology classes proven in Proposition 3.4. Together with the fact that the non-large classes form a vector subspace (and hence contain the class 0 in each degree), this shows indeed that enlargeable manifolds and more generally manifolds with (coarsely) hyperspherical, (coarsely) hypereuclidean or macroscopically large universal covers are essential. In Section 4 we will illustrate by examples that the subspaces H sm(p) are in general different from zero and may even depend on the specific largeness property P. In particular we will prove the following consequence of Theorem

3 Theorem 1.4. For all n 4 there are enlargeable (hence essential) manifolds of dimension n whose universal covers are neither (coarsely) hyperspherical, (coarsely) hypereuclidean nor macroscopically large. Gromov asked in [11, page 113] whether universal covers of essential manifolds were always hyperspherical. Theorem 1.4 gives a negative answer. It also shows the remarkeable fact that enlargeable manifolds do not always have hyperspherical universal covers. This provides a late justification for the organisation of an argument in [18]: In that paper the proof that enlargeable manifolds are Baum-Connes essential was much easier for manifolds with hyperspherical universal covers (these were called universally enlargeable in [18]) than for general enlargeable manifolds. Now we see that the general case cannot be reduced to the case of manifolds with hyperspherical universal covers. By a refinement of our methods we also get the following result, showing that even the more flexible property of enlargeability is not always implied by essentialness. Theorem 1.5. For all n 4, there are n-dimensional closed manifolds which are essential, but not enlargeable. Because arbitrary covers of these manifolds need to be controlled, the proof of this result is technically much harder than the one of Theorem 1.4. Our argument makes essential use of the Higman 4-group [21]. These examples are interesting because enlargeability is the most flexible of the largeness properties considered in this paper in terms of which coverings of the given manifold can be used. In this respect they point to a principal limitation of the use of largeness of closed manifolds for proving the strong Novikov conjecture in full generality, cf. [18, 19]. We remark however that none of our examples appearing in Theorem 1.4 or 1.5 is aspherical. In the case of aspherical manifolds there is apparently room to use metric largeness properties in order to prove general theorems on the non-existence of positive scalar curvature metrics and related properties, notably if the fundamental group has finite asymptotic dimension [6, 7]. The characterization of metric largeness by certain subspaces of group homology has been remarked before in the context of vanishing simplicial volume [9, Section 3.1] and in some more restricted setting in the context of positive scalar curvature metrics on high-dimensional manifolds with non-spin universal covers [24]. Recently [2] the first author of the present article showed that the systolic constant, the minimal volume entropy, and the spherical volume of closed manifolds only depend on the image of their fundamental classes in the integral homology of their fundamental groups under the classifying maps of their universal covers. In the definition of enlargeability the maps M ε S n are assumed to contract distances. If they are only required to contract volumes of k-dimensional submanifolds, then M is called k- enlargeable. In the case k = 2, this property is also called area-enlargeability [15]. Relying on index theory it was shown in [20] that area-enlargeable manifolds are essential. In Section 5 of the paper at hand we will prove the following more general statement by methods similar to those employed in Section 3. Theorem 1.6. Let M be a closed oriented n-dimensional manifold. If M is k-enlargeable and satisfies π i (M) = 0 for 2 i k 1, then M is essential. In particular, area-enlargeable manifolds are essential. 3

4 For k 2 the condition on the homotopy groups is to be understood as empty. Note that for k > 2 the condition is in fact necessary: Let M be an enlargeable manifold. Then the product M S 2 is 3-enlargeable, but the classifying map M S 2 Bπ 1 (M) sends the fundamental class to zero, i. e. M S 2 is not essential. This is in accordance with Theorem 1.6 as π 2 (M S 2 ) 0. For k n + 1 the k-dimensional volume of any subset of S n is zero, of course. Therefore, in this case the assumption of k-enlargeability in Theorem 1.6 can be dropped and the remaining nontrivial requirement is π i (M) = 0 for 2 i k 1. The inequality k n+1 and the Hurewicz theorem then imply that all homotopy groups of the universal cover of M vanish. In other words, Theorem 1.6 includes the well known statement that aspherical manifolds are essential. Hence the conditions in Theorem 1.6 interpolate between two extreme cases: enlargeable and area-enlargeable manifolds on the one side and aspherical ones on the other side. It was shown in [15] that area-enlargeable spin manifolds do not carry metrics of positive scalar curvature and it has been conjectured that the conclusion is valid for all aspherical manifolds. In this respect it seems reasonable to conjecture that the conditions in Theorem 1.6 are also obstructions to the existence of positive scalar curvature metrics. In fact the strong Novikov conjecture implies that essential spin manifolds do not admit positive scalar curvature metrics, cf. [23]. Concerning homological invariance, k-enlargeability for k 2 seems to bahave less favourably than the other largeness properties considered in our work. However, we will not pursue this question further in this paper. Theorem 1.6 is related in spirit to Theorem 2.5 in [4], which deals with functorial properties of hyperbolic cohomology classes. This article is based in part on a chapter of the first author s thesis [3]. In particular, homological invariance of largeness properties (which is part of Theorem 1.3 of the present article) and Theorem 1.6 were first proved there. The most important novelties of the present paper are a more systematic treatment of large homology classes in Section 3 and - based on that - the construction of interesting examples of enlargable manifolds without hyperspherical universal covers (Theorem 1.4) and of essential manifolds that are not enlargeable (Theorem 1.5). Acknowledgements: The first author would like to thank his thesis advisor D. Kotschick for continuous support and encouragement. Both authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft, useful comments by D. Kotschick concerning a preliminary version of this paper and numerous helpful remarks by the referee. 2. LARGE RIEMANNIAN MANIFOLDS In this section we will recall classical notions of metric largeness for Riemannian manifolds, most of which were first formulated by Gromov, see for example [11, 12, 15], and also [5, 17]. They include the properties of being k-hypereuclidean and k-hyperspherical, see Definition 2.2, being k-enlargeable, see Definition 2.3, having infinite filling radius, see Definition 2.5 and being macroscopically large, see Definition 2.7. In Proposition 2.10 we will characterise hypersphericy in terms of the existence of a Lipschitz map of nonzero degree to the balloon space, which was introduced in [18]. Upon passing from Lipschitz to large scale Lipschitz maps this allows us to define coarsely hyperspherical manifolds, a notion similar to coarsely hypereuclidean manifolds, which by definition admit coarse maps to Euclidean space of nonzero degree. Furthermore we will discuss several implications between these largeness properties. In particular, in Proposition 2.8 we will show that the classes of macroscopically large 4

5 manifolds and of manifolds with infinite filling radii coincide. This is remarkable because it relates a coarse (i. e. quasi-isometric) property to a bi-lipschitz one. Let f : (M, g M ) (N, g N ) be a smooth map between (not necessarily compact) Riemannian manifolds, and let k be a positive integer. Definition 2.1. The k-dilation of f is defined as dil k (f) := sup Λ k D p f R { }, p M the supremum of the norms of the k-fold exterior product of the differential Df. Said differently, the k-dilation is the smallest number ε such that for any k-dimensional submanifold A M the k-dimensional volume Vol k (f(a)) of the image f(a) N is bounded by ε Vol k (A). The 1-dilation is the smallest Lipschitz constant for f. Let p M be a point, and let n be the dimension of M. Denote by λ 1... λ n > 0 the eigenvalues of the Gram matrix of the pullback (D p f) (g N ) f(p) with respect to (g M ) p. Then Λ k D p f 2 = λ 1... λ k. Therefore, the inequality dil l (f) 1/l dil k (f) 1/k holds for all l k. Let (V, g) be a complete orientable Riemannian manifold of dimension n. A choice of orientation for V defines a fundamental class [V ] Hn lf (V ; Z) in locally finite homology. In this context the mapping degree is well-defined for proper maps to oriented manifolds and for maps to closed oriented manifolds Z that are almost proper, i.e. constant outside a compact set. This can be made rigorous by adding an infinite whisker to Z, extending the given almost proper map to a proper map with target Z whisker, and observing that H lf (Z whisker) = H (Z). In the following, we equip Euclidean spaces and unit spheres with their standard metrics. Definition 2.2. We call (V, g) k-hypereuclidean if there is a proper map f : V R n of nonzero degree and of finite k-dilation. It is called k-hyperspherical if for every ε > 0 there is an almost proper map f ε : V S n of nonzero degree such that dil k (f ε ) ε. For k = 1 we will omit the number, and for k = 2 we will speak of area-hypereuclidean and area-hyperspherical manifolds. By the above inequality, every k-hypereuclidean or k-hyperspherical manifold is also l-hypereuclidean respectively l-hyperspherical for any l k. Since R n is obviously hyperspherical, any k-hypereuclidean manifold is also k-hyperspherical. Note also that both notions depend only on the bi-lipschitz type of the metric g. Closely related is the notion of enlargeability. It was introduced by Gromov and Lawson in [14] and in the following more general form in [15]. Definition 2.3. An orientable n-dimensional manifold V is called k-enlargeable if for every complete Riemannian metric g on V and every ε > 0 there is a Riemannian cover V ε of V and an almost proper map f ε : V ε S n 5

6 of nonzero degree such that dil k (f ε ) ε. As before, we will omit the number k in the case k = 1 and speak of area-enlargeable manifolds in the case k = 2. If V is closed, then all Riemannian metrics on V are bi-lipschitz to each other and it is enough that V satisfies the above condition with respect to one Riemannian metric. The significance of the notion of enlargeability is demonstrated by the following theorem, see [15, Theorem 6.1]. Theorem 2.4. If V is area-enlargeable and the covers V ε in Definition 2.3 may be chosen spin, then V does not carry a complete Riemannian metric of uniformly positive scalar curvature. Next, we will revisit the notion of filling radius. Recall that every Riemannian metric g on V induces a path metric d g on V. Denote by L (V ) the vector space of all functions V R with the uniform norm. This is not a norm proper since it may take infinite values. Therefore the induced metric is not an actual metric. Nevertheless, the Kuratowski embedding ι g : (V, d g ) L (V ), v d g (v, ) is an isometric embedding by the triangle inequality. One could replace L (V ) by its affine subspace L (V ) b that is parallel to the Banach space of all bounded functions on V and contains the distance function d g (v, ) for some v V. Then the image of the Kuratowski embedding is contained in L (V ) b, and the norm induces an actual metric on L (V ) b. Since all points of L (V ) outside of this affine subspace are already infinitely far away from it, this would not change the following definition. Definition 2.5. The filling radius of (V, g) is defined as FillRad(V, g) := inf{r > 0 ι g [V ] = 0 H lf n (U r (ι g V ); Q)} where U r (ι g V ) L (V ) denotes the open r-neighborhood of the image ι g V L (V ). If the set on the right hand side is empty, we say that (V, g) has infinite filling radius. Note that for closed manifolds L (V ) b is the vector space of all bounded functions on V and the above definition of the filling radius coincides with the usual definition (see [10], Section 1). For noncompact manifolds the filling radius need not be finite. For instance the filling radius of the Euclidean space is infinite. It follows from the definition that the property of having infinite filling radius depends only on the bi-lipschitz type of the metric g. We recall the following implication that was shown in [11] (see also [5]). Proposition 2.6. If (V, g) is hyperspherical, then its filling radius is infinite. We have seen that hypereuclidean manifolds are hyperspherical and that hyperspherical manifolds have infinite filling radius. It is not known whether these implications are equivalences or not. In [8] Gong and Yu used coarse algebraic topology to define another notion of largeness, which is thought to be closely related to Gromov s definitions. In fact we will show that it is equivalent to the property of having infinite filling radius. First we will recall some basics on coarse geometry. For more details we refer to Roe s book [22], in particular to Chapter 5 on coarse algebraic topology. 6

7 Let X be a metric space. A cover U of X is called uniform if the diameters of its members are uniformly bounded and if every bounded set in X meets only finitely many members of U. A familiy (U) i I of uniform covers is called anti-čech system if for every r > 0 there exists a cover U i with Lebesgue number at least r. The nerve of a cover U will be denoted by U. It is the simplicial complex whose simplices are finite subsets of U with nonempty intersection in X. In particular the set of vertices is equal to U. The nerve of a uniform cover is locally finite. If U and V are two uniform covers such that the Lebesgue number of V is bigger than the uniform bound on the diameters of the sets of U, we write U V. In this way the set of uniform covers of X becomes directed. If U V, then there is a proper simplicial map U V mapping each vertex U U to some vertex V V that contains U. The proper homotopy class of this map is uniquely determined. If X is proper (i.e. bounded subsets are precompact) then anti-čech systems alway exist. Given an anti-čech system one defines the coarse homology of X as HX k (X; Q) := lim H lf k ( U i ; Q). This is independent of the choice of the anti-čech system. For proper X and for any uniform cover U there is a proper map X U that sends each point x X to a point in the simplex spanned by those U U that contain x. Moreover, the proper homotopy class of such a map is uniquely determined. Therefore, one gets an induced homomorphism c : H lf k (X; Q) HX k (X; Q), which will be called the character homomorphism of X. Definition 2.7 ([8]). A complete oriented n-dimensional Riemannian manifold V is called macroscopically large, if c[v ] 0 HX n (V ; Q). Note that this property depends only on the quasi-isometry class of the metric. We will show that macroscopic largeness is equivalent to having infinite filling radius. This proves that the property of having infinite filling radius depends only on the quasi-isometry class of the Riemannian metric. Note that the quasi-isometry class strictly includes the bi-lipschitz class. It is not known whether hypereuclideaness or hypersphericity are also invariant under quasi-isometries. Proposition 2.8. Let V be a complete orientable Riemannian manifold. Then V is macroscopically large if and only if its filling radius is infinite. For the proof we will need the notion of a coarse map. The general definition is a bit involved. But recall from [22, Section 1.3] that a map f : X Y from a path metric space to a metric space is coarse if and only if it is large scale Lipschitz and (metrically) proper. Proof. Identify V with its image under the Kuratowski embedding, and let n be the dimension of V. First assume that FillRad(V, g) < r for some finite r. Then there is a locally finite complex X U r (V ) containing V such that [V ] = 0 Hn lf (X; Q). Moreover, the inclusion V X is a coarse equivalence since the coarse map that assigns to a point x X a point v V with 7

8 d(x, v) r is an inverse. The commutative diagram H lf n (V ; Q) c HX n (V ; Q) H lf n (X; Q) = HX n (X; Q) c shows that c[v ] = 0 HX n (V ; Q), i. e. (V, g) is not macroscopically large. (Note that U r (V ) is also coarsely equivalent to V but it is not proper. Therefore, it is not clear whether it admits a character homomorphism.) To prove the converse implication, assume that (V, g) is not macroscopically large. By the definition of the direct limit there is a uniform cover U of V such that φ [V ] = 0 Hn lf ( U ; Q) where φ : V U is a proper map that sends each point v V to a point in the simplex spanned by those U U that contain v. Let r > 0 be an upper bound on the diameters of the sets of U. Define a map ψ : U L (V ) by sending each vertex U U to some point ψ(u) U V and by extending this linearly over each simplex of the nerve. Let p be a point in U. It may be written as p = λ i U i with λ i = 1, λ i > 0, and U i U such that U i. Then ψ(p) = λ i ψ(u i ) and d(ψ(p), ψ(u 1 )) = λ i ψ(u i ) ψ(u 1 ) λ i ψ(u i ) ψ(u 1 ) 2r since U i U 1 for all i. This shows that the image of ψ lies in the 2r-neighborhood of V in L (V ). Hence (ψ φ) [V ] = 0 H lf n (U 2r (V ); Q). Let v V be a point. Say v lies in the sets U 1,..., U m U and in no other set of U. Then φ(v) = m i=1 λ iu i for some λ i 0 with λ i = 1. Therefore d(ψ(φ(v)), v) = λ i ψ(u i ) v λ i ψ(u i ) v r since v U i for all i = 1,..., m. Thus the linear homotopy from the inclusion V L (V ) to ψ φ is proper and lies entirely in U r (V ). We conclude [V ] = (ψ φ) [V ] H lf n (U 2r (V ); Q), and consequently [V ] = 0 H lf n (U 2r (V ); Q), hence FillRad(V, g) 2r <. Propositions 2.6 and 2.8 show that complete hyperspherical manifolds are macroscopically large. If the given hyperspherical manifold is the universal cover of a closed manifold, this is proved directly in [18, Proposition 3.1] using the balloon space B n. This path metric space is defined as a real half-line [0, ) with an n-dimensional round sphere Si n of radius i attached at each positive integer i [0, ). 8

9 Proposition 2.9 ([18], Proposition 2.2). The n-dimensional coarse homology of the balloon space is given by ( ) ( ) HX n (B n ; Q) = Q / Q. i=1 Moreover, for the locally finite homology we have Hn lf (B n ; Q) = i=1 Q, and the character homomorphism c : Hn lf (B n ; Q) HX n (B n ; Q) is the canonical projection. Using this computation we obtain the following characterization of hyperspherical manifolds. Proposition A complete oriented Riemannian manifold V of dimension n is hyperspherical, if and only if there exists a proper Lipschitz map f : V B n such that f [V ] 0 HX n (B n ; Q). Proof. First, assume that V is hyperspherical. We will construct a sequence of closed balls i=1 = B 0 B 1 B 2... V that exhausts V and a sequence of 1-Lipschitz maps f i : B i \ B i 1 Si n [i, i + 1] B n such that f i ( B i 1 ) = i, f i ( B i ) = i + 1, and such that f i is of nonzero degree as a map to Si n. Assume that the balls and maps have been constructed up to index i 1. Let SR n be the round sphere of radius R with a large R which will be specified later. Choose a 1-Lipschitz map f i : V SR n that is constant outside a compact set K i and that is of nonzero degree. Without loss of generality, we may assume that B i 1 K i and that f i(b i 1 ) and the point f i(v \ K i ) avoid a ball of radius πi in SR n (choose for instance R 2i + r/π where r is the radius of B i 1). Let g i : SR n Sn i be a nonexpanding map that contracts everything outside this ball of radius πi to the base point of Si n. Choose a ball B i V such that K i B i and such that d( B i, K i ) 1. Define f i as follows: { g i f f i (v) := i(v) for v B i \ B i 1 and d(v, B i ) 1 i + 1 d(v, B i ) for v B i, d(v, B i ) 1 Then f i has the asserted properties. The collection of the maps f i defines a proper 1-Lipschitz map f : V B n such that every entry of f [V ] Hn lf (B n ; Q) = i=1 Q is nonzero. In particular f [V ] 0 HX n (B n ; Q) as required. For the converse, let f : V B n be a proper Lipschitz map such that f [V ] 0 HX n (B n ; Q). Let ε > 0, and choose an integer i dil 1 (f)/ε such that the i-th entry of f [V ] Hn lf (B n ; Q) = i=1 Q is not zero. This is possible since by assumption there are infinitely many nonvanishing entries. Let f ε be the composition of f with the canonical quotient map from B n to the i-th sphere Si n and the dilation from this sphere of radius i to the unit sphere. Then f ε is constant outside a compact set, has nonzero degree, and its dilation is given by dil 1 (f)/i ε. This proves that V is indeed hyperspherical. In [18, Proposition 3.1] it is shown that hyperspherical universal covers of closed Riemannian manifolds admit proper Lipschitz maps to the balloon space sending the locally finite fundamental class of the universal covers to nonzero classes. The corresponding implication in our Proposition 2.10 is slightly more general in that it is not assumed that the metric on V is invariant under a cocompact group action. 9

10 Definition A complete oriented Riemannian manifold V of dimension n is called coarsely hypereuclidean, if there is a coarse map f : V R n such that f [V ] 0 HX n (R n ; Q) = Q. It is called coarsely hyperspherical if there is a coarse map f : V B n to the balloon space such that f [V ] 0 HX n (B n ; Q). These two notions depend only on the quasi-isometry class of the metric on V. The following diagram summarizes the known implications between some of the largeness properties on a complete Riemannian manifold discussed in this section. hypereuclidean coarsely hypereuclidean hyperspherical coarsely hyperspherical infinite Prop.2.8 macroscopically filling radius large Prop.2.6 As H lf n (R n ; Q) = HX n (R n ; Q) = Q, hypereuclidean manifolds are coarsely hypereuclidean, and Proposition 2.10 implies that hyperspherical manifolds are coarsely hyperspherical. This explains the two upper horizontal arrows. Moreover, the proof of Proposition 2.10 shows the existence of a coarse map R n B n sending the coarse fundamental class of R n to a non-zero class in HX n (B n ; Q). This implies the upper vertical arrow on the right. The lower right vertical implication follows by the very definition of macroscopic largeness. Apart from the lower horizontal arrow it is not known if any of the implications is an equivalence. We also remark that the properties on the left-hand side are invariants of the bi-lipschitz class of the given metric, the ones on the right-hand side of its quasi-isometry class. 3. LARGENESS IN HOMOLOGY In this section we shall formulate the concept of largeness for rational homology classes in simplicial complexes with finitely generated fundamental groups. We will then prove Theorem 1.3. In this section, the term large is a placeholder for one of the properties of being enlargeable or having a universal cover which is (coarsely) hyperspherical, (coarsely) hypereuclidean or macroscopically large. The method of extending differential-geometric concepts from smooth manifolds to more general spaces like simplicial complexes has occured at other places in the literature in similar contexts, see e. g. [2]. We equip the n-dimensional simplex n with the metric induced by the standard embedding into R n+1. Recall from [13, Chapter 1] that each connected simplicial complex comes with a canonical path metric restricting to this standard metric on each simplex. Furthermore, if p : X Y is a covering map of path connected topological spaces and Y is equipped with a path metric, then there is a unique path metric on X so that p is a local isometry. If Y is a simplicial complex with 10

11 the canonical path metric and X carries the induced simplicial structure, then this is the canonical path metric on X. A connected subcomplex S of a connected simplicial complex X is called π 1 -surjective, if the inclusion induces a surjection of fundamental groups and we say that S carries a homology class c H (X; Q), if c is in the image of the map in homology induced by the inclusion. If p : X X is a (not necessarily connected) cover of a simplicial complex X and c H n (X; Q) is a simplicial homology class, the transfer p! (c) Hn lf (X; Q) is represented by the formal sum of all preimages of simplices in a chain representative of c, with appropriate coefficients. Definition 3.1. Let X be a connected simplicial complex with finitely generated fundamental group and let c H n (X; Q) be a (simplicial) homology class. Choose a finite connected π 1 - surjective subcomplex S X carrying c. (This subcomplex exists, because π 1 (X) is finitely generated.) The class c H n (X; Q) is called enlargeable, if the following holds: Let ε > 0. Then there is a connected cover p : X X and an almost proper ε-contracting map f ε : S S n which sends the class p! (c) to a nonzero class in H n (S n ; Q). Here S := p 1 (S) (which is connected as S is π 1 -surjective) is equipped with the canonical path metric. The class c is called (coarsely) hypereuclidean, (coarsely) hyperspherical, respectively macroscopically large if the complex S = p 1 (S) associated to the universal cover p : X X together with the transfer class p! (c) enjoys the according property. It is important to work with π 1 -surjective subcomplexes S, because then the covers S are connected and hence equipped with canonical path metrics. We could have replaced the conditions in Definition 3.1 by first representing the homology class in question as the image of the fundamental class under a map φ : M X from a closed oriented n-dimensional manifold M to X and requiring that M or appropriate covers thereof share the corresponding largeness property. This indeed works if φ induces a surjection in π 1. We preferred the above definition because it applies as well to homology classes which are not representable by closed manifolds (which is relevant in low dimensions), works for fundamental groups which are not necessarily finitely presented and last but not least shows that metric largeness properties are not linked to differential-topological properties of manifolds, but only to metric properties of simplicial complexes. Of course this will not prevent us from applying the results of the present section mainly to the classical case of closed manifolds later on. We remark that a map defined on a connected simplicial complex X (with the path metric) to a metric space is ε-contracting if and only if this holds for the restriction of the map to each simplex in X. We need to show that Definition 3.1 does not depend on the choice of S. This is in fact the main technical argument in this section and we will explain it first in detail for enlargeable classes. We start with the following basic extension lemma. Lemma 3.2. Let k, n 1 be natural numbers and equip the disk D k and its boundary D k with fixed, but arbitrary piecewise smooth Riemannian metrics (which need not be related). Then there are positive constants δ and C which depend only on the chosen metrics on D k and D k and on n, so that the following holds: Let 0 < ɛ < δ and let f : D k S n be a piecewise smooth ε- contracting map. Then f can be extended to a piecewise smooth (C ε)-contracting map D k S n. 11

12 Proof. Because any two piecewise smooth metrics on D k are in bi-lipschitz correspondence, it is enough to treat the case when the metric on D k is given in polar coordinates by dr 2 + r 2 g where g is the given piecewise smooth Riemannian metric on D k. We can choose a δ > 0 so that for any 0 < ε < δ and any ε-contracting piecewise smooth map f : D k S n, the image of f is contained in a closed hemisphere D S n. Using polar coordinates, we identify the hemisphere with the unit ball D n R n by a diffeomorphism ω : D n D. This diffeomorphism is bi-lipschitz with Lipschitz constants independent of f. In particular there is a constant C, independent of ɛ an f, so that ω 1 f : D k D n is (C ɛ)-contracting. This implies that the diameter of the image of ω 1 f is bounded above by diam( D k ) C ε. Now map the midpoint P of D k to some point contained in the image of ω 1 f and extend ω 1 f to D k linearly along the radial lines joining P. This extended map is piecewise smooth and using polar coordinates on D k for computing the lengths of piecewise smooth curves in D k (and the fact that the metric on D k has the special form described above), this extended map is C ε-contracting with a constant C which is independent of ε and f. Upon composing this map with the Lipschitz map ω, the claim of the lemma follows. Returning to Definition 3.1 let S S be a smaller finite connected π 1 -surjective subcomplex which carries c. If one of the properties described in Definition 3.1 holds for S, then it holds as well for S by the naturality of p! and because the lifted inclusion S S is 1-Lipschitz for any connected cover X X. Now let S, S X be two finite connected π 1 -surjective subcomplexes carrying c. Then there is a third finite connected π 1 -surjective subcomplex T X carrying c and containing S and S. Hence it remains to show that in Definition 3.1 we may pass from S to a larger finite connected π 1 -surjective subcomplex T of X. Let ε > 0. By assumption there is a connected cover p : X X and an almost proper ε- contracting map f ε : S S n satisfying (f ε ) (p! (c)) 0. We will show that if ε is small enough, then f ε can be extended to a (C ε)-contracting almost proper map T S n where C > 0 is a constant which depends only on S and T, but not on ε. The proof is by induction on the k-skeleta T (k) of T, where 0 k dim T. However, the start of the induction is a bit involved, because we need to treat the cases k = 0, 1 simultanously. Let us first assume that T \ S contains just one vertex v. Let V T be the set of lifts of v. For each v V, let F (v) S be the set of all vertices having a common edge with v. Note that because T is connected and locally finite, the set F (v) is nonempty and finite. Furthermore, diam F (v) (measured with respect to the path metric on S) is independent of v V. Let F (ṽ) S be the subset defined in an analogous fashion as F (v) but with S replaced by the universal cover S S (and v by a point ṽ S over v) and set d := diam F (ṽ) measured with respect to the path metric on S. Then d is independent of the choice of ṽ and ε and furthermore diam F (v) d. Let e T be a fixed edge connecting v with a vertex in S. For v V, we set f ε (v) := f ε (v 1 (e(v))) where e(v) is the uniqe lift of e containing v and v 1 (e(v)) is the vertex of this lift different from v. The extension f ε : S {v} S n defined in this way satisfies d(f ε (v 0 ), f ε (v 1 )) max{d, 1} ε, if v 0 and v 1 are the vertices in some 1-simplex of S T (1) = p 1 (S T (1) ). Next, assuming that 12

13 max{d, 1} ε < δ 1, we extend f ε to a (max{d, 1} C 1 ε)-contracting map S T (1) S n using Lemma 3.2. Here, δ 1 and C 1 are given by Lemma 3.2 and depend only on S and T. If T \S contains more than one vertex, we apply this process inductively, where in each induction step, we pick a vertex in T which has a common edge with some vertex in the subcomplex where f ε has already been defined (note that in each step, this subcomplex is connected). In this way, we get (for small enough ε) a (C 1 ε)-contracting, almost proper extension S T 1 S n of f ε where C 1 > 0 is an appropriate constant which just depends on S and T. Now, if for k 3, we have a (C 1 C 2... C k 1 ε)-contracting (respectively, if k = 2, a (C 1 ε)-contracting) almost proper extension S T (k 1) S n of f ε, we extend this map to a (C 1... C k ε)-contracting almost proper map S T (k) S n. This is possible by Lemma 3.2, if ε is small enough (where the smallness just depends on S and T ). A similar argument works for hypereuclidean and hyperspherical classes. If we are dealing with coarse largeness properties, the preceding argument can be replaced by the simple observation that the inclusion S T is a coarse equivalence, where S = p 1 (S) is the preimage of S under the universal covering map p : X X and similarly for T. As a further consequence of our argument we notice the following slightly different characterisation of large classes which will be important for one of our constructions in the next section. Proposition 3.3. Let X be a connected simplicial complex with finitely generated fundamental group and let c H n (X; Q) be a homology class. Then c is enlargeable, if and only if the following holds: Choose a finite connected π 1 -surjective subcomplex S X carrying c. Let C S be a finite subcomplex carrying c which is not necessarily connected or π 1 -surjective. Then, for any ε > 0, there is a connected cover p : X X and an ε-contracting almost proper map C S n mapping p! (c) to a nonzero class. Here, C := p 1 (C) S = p 1 (S) is equipped with the restriction of the canonical path metric on S. Proof. It is easy to see that the given property is necessary for the enlargeability of c. Conversely, if this condition is satisfied by C we apply the preceding argument in order to show that it is also satisfied by the larger subcomplex S. The only difference is that in the beginning of the induction (i.e. for k = 0, 1) we work with the restrictions of the path metrics from S to C and from S to C (for the defintition of d). Here C := ψ 1 (C), where ψ : X X is the universal cover. We remark that analogues of Proposition 3.3 hold for (coarsely) hypereuclidean, (coarsely) hyperspherical or macroscopically large classes. Next we study functorial properties of large homology classes. Proposition 3.4. Let X and Y be connected simplicial complexes with finitely generated fundamental groups and let φ : X Y be a continuous (not necessarily simplicial) map. Then the following implications hold: If φ induces a surjection of fundamental groups and φ (c) is enlargeable, then c is enlargeable. If φ induces an isomorphism of fundamental groups and c is enlargeable, then also φ (c) is enlargeable. 13

14 If we are dealing with (coarsely) hyperspherical, (coarsely) hypereuclidean or macroscopically large classes and if we assume that φ induces an isomorphism of fundamental groups, then c is large in the respective sense if and only if φ (c) is. Proof. First, assume that φ (c) is enlargeable and φ is surjective on π 1. Let S X be a finite connected π 1 -surjective subcomplex carrying c. Then φ(s) is contained in a finite connected π 1 - surjective subcomplex T Y which carries φ (c). Because S and T are compact, the map φ : S T is Lipschitz with Lipschitz constant λ, say. Let ε > 0 and choose a connected cover p Y : Y Y together with an almost proper ε- contracting map T S n mapping (p Y )! (c) to a nonzero class. Let p X : X X be the pull back of this cover under φ. Because φ is surjective on π 1, X is connected and we get a map of covering spaces p X S φ T S φ T which restricts to a bijection on each fibre. In particular, the map φ is proper and λ-lipschitz (as usual, S is equipped with the canonical path metric). Hence, if f ε : T S n is an almost proper ε-contracting map, then f ε φ is almost proper, λ ε contracting and maps (p X )! (c) to a nonzero class. Now assume that c is large and φ induces an isomorphism of fundamental groups. By the first part of this proof, we can replace Y by a homotopy equivalent complex and hence we may assume that φ is an inclusion. Let S X be a finite connected subcomplex carrying c. Then S is also a subcomplex of Y and it carries φ (c). Because φ induces an isomorphism on π 1, each connected cover of X can be written as the restriction of a connected cover of Y. This shows that φ (c) is also enlargeable. Again, the other largeness properties are treated in a similar manner. Proposition 3.4 implies the important fact that the large classes form a well-defined subset of H (Γ; Q) = H (BΓ; Q) for each finitely generated group Γ in the following sense: For each simplicial model X of BΓ, the large classes form a subset of H (X; Q) and if X and Y are two models of BΓ and X Y is the (up to homotopy unique) homotopy equivalence inducing the identity on π 1 = Γ, then the induced map in homology identifies the large classes in H (X; Q) and H (Y ; Q). The next corollary states homological invariance of classical largeness properties. Corollary 3.5. Let M be a closed oriented manifold of dimension n. Then M is large if and only if φ [M] H n (Bπ 1 (M); Q) is large. Proof. This is implied by Proposition 3.4, because M is large, if and only if [M] H n (M; Q) is large. The next theorem indicates that the subset of non-large classes should actually be our main concern. Theorem 3.6. Let X be a connected simplicial complex with finitely generated fundamental group. Then the non-large homology classes in H n (X; Q) form a rational vector subspace. 14 p Y

15 Proof. The class 0 H n (X; Q) is not large: Take any connected finite π 1 -surjective subcomplex S X. Clearly, S carries 0. By a direct application of Definition 3.1 it follows that 0 is not large. It is obvious that if c H n (X; Q) is not large, then no rational multiple of c is large. In order to show that the subset of non-large classes is closed under addition, we need to show the following: Let c, d H n (X; Q) and assume that c + d is large. Then one of c and d must also be large. For a proof, let S X be a connected finite π 1 -surjective subcomplex carrying c and d. Then S also carries c+d. Assume that c+d is enlargeable (the other largeness properties are easier and left to the reader). Let ε := 1 for a natural number k 1. Because c + d is enlargeable, there k is a connected cover p : X X and an almost proper ε-contracting map f ε : S S n mapping p! (c + d) to a nonzero class in H n (S n ; Q). This can hold only if either p! (c) or p! (d) is mapped to a nonzero class. Hence, for infinitely many k, either p! (c) or p! (d) is mapped to a nonzero class (for appropriate covers p : X X) and consequently either c or d is enlargeable. Definition 3.7. If one largeness property P is chosen and X is a connected simplicial complex with finitely generated fundamental group, we denote the rational vector subspace of H (X; Q) consisting of classes which are not large with respect to P by H sm(p) (X; Q). This is the small homology of X with respect to P and depends a priori on the given largeness property P. Theorem 3.6 implies that 0 Hn sm(p) (BΓ; Q) for each finitely generated group and each largeness property P. Together with Corollary 3.5 this shows again that large manifolds are essential. The results of this section leave it as a central problem to determine H sm(p) (BΓ; Q) for different largeness properties and finitely generated groups Γ. This will be the topic of the next section. 4. EXAMPLES AND APPLICATIONS The following theorem illustrates the usefulness of our systemtic approach to large homology in Section 3. Theorem 4.1. The small homology of BZ k = T k, k 1, is calculated as follows. If P denotes enlargeability we have Hn sm(p) (T k ; Q) = 0 for all n 1. If P denotes (coarse) hypereuclideaness, (coarse) hypersphericity and macroscopic largeness we have H sm(p) n (T k ; Q) = { 0 for n = k H n (T k ; Q) for 1 n < k. Proof. We equip T k = R k /Z k with the metric induced from R k. Let 0 c H n (T k ; Q). We write c = α i1,...,i n t i1,...,i n 1 i 1 <i 2 <...<i n k where t i1,...,i n H n (T k ; Z) is represented by the embedding T n T k, (ξ 1,..., ξ n ) (1,..., 1, ξ 1, 1,..., 1, ξ n, 1,..., 1) of the n-torus into T k = (S 1 ) k, where ξ ν is put at the i ν th entry, and each α i1,...,i n loss of generality we may assume that α 1,...,n Q. Without

16 Let p : R n T k n T k be the cover associated to the subgroup Z k n = 0 Z k n Z k given by the last k n coordinates. For ε > 0, let f : R n S n be an ε-contracting almost proper map of nonzero degree. Then the composition is ε-contracting, almost proper and f ε : R n T n k pr R n R n f S n (f ε ) (p! (t i1,...,i n )) 0 H n (S n ; Q) if (i 1,..., i n ) = (1, 2,..., n) and is zero otherwise. Hence, (f ε ) (p! (c)) 0. This shows that c is enlargeable. Concerning the other largeness properties it is clear that H sm(p) k (T k ; Q) = 0, because the universal cover R k of T k shares all of the mentioned largeness properties. Futhermore, notice that the transfer maps each non-zero class in H k (T k ; Q) to a rational multiple of the locally finite fundamental class of R k. Now let 1 n < k and let c H n (T k ; Q). The full space T k carries c and is π 1 -surjective. However, if n < k, then Poincaré duality implies H lf n (R k ; Q) = H k n (R k ; Q) = 0 so that under the universal cover p : R k T k, the transfer p! (c) Hn lf (R k ; Q) is equal to zero. This shows that c cannot be large. Combined with Corollary 3.5 this calculation has the following interesting consequence. Theorem 4.2. Let M be a closed orientable manifold of dimension n and with fundamental group Z k, where 1 n < k. Then M is enlargeable if and only if it is essential. However, the universal cover of M is never (coarsely) hypereuclidean, (coarsely) hyperspherical or macroscopically large. For 4 n < k we can construct essential n-dimensional manifolds M with fundamental group Z k as follows. Start with the oriented connected sum C := T n ( k n (S 1 S n 1 ) ) of an n-torus with k n copies of S 1 S n 1. This manifold has fundamental group π 1 (C) = Z n ( k n Z), the free product of Z n with k n copies of Z. Let C Bπ 1 (C) be the classifying map of the universal cover and consider the composition φ : C Bπ 1 (C) BZ k induced by the abelianization Z n ( k n Z) Z k. Then φ sends the fundamental class of C to a non-zero class in H n (T k ; Z). Moreover it induces a surjective map of fundamental groups π 1 (C) π 1 (BZ k ) = Z k whose kernel can be killed by oriented surgeries in C (here we use the assumption n 4). The resulting manifold M has the stated properties. Together with Theorem 4.2 this completes the proof of Theorem 1.4. We will now describe a refined construction to obtain essential manifolds which are not enlargeable. As we have to deal with arbitrary covers of the manifolds in question, we need to recall some facts from covering space theory. Let X be a path connected space and let S X be a path connected subspace. We choose a base point x in the universal cover X of X which lies over S and compute all fundamental groups with respect to x and its images in the different covers of X. Let G = π 1 (X), let H G be a subgroup and let λ : π 1 (S) π 1 (X) be the map induced by the inclusion S X. We denote the 16

17 image of this map by Σ π 1 (X). Let p : X X be the cover associated to H. Recall that G acts on X from the left in a canonical way. In view of Proposition 3.3 we collect some information on the components of S := p 1 (S). The projection X X is denoted by ψ. If g 1, g 2 G, then g 1 x and g 2 x are mapped to the same component of S if and only if there are elements σ Σ and h H with g 1 σ = hg 2. Hence the components of S are in one to one correspondence with double cosets H\G/Σ. If P S is one component, then there exists a g G so that ψ(g x) P. With respect to this base point, the fundamental group of P is canonically isomorphic to λ 1 (g 1 Hg) π 1 (S). Replacing g x by hgσ x gives another point whose image lies in P, and with respect to this base point the fundamental group of P is equal to λ 1 (σ 1 g 1 Hgσ). If λ is injective, we identify π 1 (S) and Σ and then the fundamental group of the component P S with respect to the base point ψ(g x) is equal to Σ (g 1 Hg). If moreover Σ is normal in π 1 (X), this group is equal to g 1 (Σ H)g and so the fundamental groups of the components P S are mutually isomorphic. Let Hig := a, b, c, d a 1 ba = b 2, b 1 cb = c 2, c 1 dc = d 2, d 1 ad = a 2 be the Higman 4-group [21]. This is a finitely presented group with no proper subgroups of finite index. By [1] Hig is integrally acyclic, i. e. H (Hig; Z) = 0. Pick a z Hig of infinite order. We define K := Hig z Hig as the amalgamated free product of Hig with itself along z. We claim that the group K still does not possess any proper subgroups of finite index. Assume the contrary and let H < K be a proper subgroup of finite index. Then the homomorphism from K to the permutation group of K/H induced by the left translation action of K is nontrivial and has finite image. But then the push out property of the amalgamated free product implies that also Hig has a nontrivial homomorphism to a finite group, contradicting the fact that Hig has no proper subgroups of finite index. A Mayer-Vietoris argument shows that H (K; Z) = H (S 2 ; Z). Let c H 2 (K; Z) be a generator and define the group L by the central extension 1 Z L K 1 classified by c and the group N by the central extension 1 Z N Z 2 1 classified by a nontrivial generator of H 2 (Z 2 ; Z) = Z. Notice that N is the fundamental group of the closed oriented 3-manifold arising as the total space of the S 1 -bundle over T 2 with Euler number 1. This can be regarded as the quotiend Nil 3 /N of the corresponding 3-dimensional nilpotent Lie group by the cocompact lattice N. We consider Z as a central subgroup of N L via the diagonal embedding and finally set G := (N L)/Z. In the following, we will regard N as a normal subgroup of G via the inclusion N = N 1 N L G and Z as a central subgroup of G via the inclusion Z N G. The following formulates a key property of G. Theorem 4.3. Let H G be a subgroup which maps surjectively onto K under the canonical map p : G N/Z L/Z L/Z = K. 17